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Matrix Mechanics Viewpoint

In order to solve the Schrödinger equation, $\hat{H} \Psi = E \Psi$, via Heisenberg's matrix mechanics, one must first select a suitable linear vector space of N-particle functions into which the problem can be expanded. Slater determinants or CSFs are a natural choice of N-particle functions for a system of electrons because they conform to most of the restrictions imposed upon a wave function describing a system of fermions. The use of Slater determinants can also be motivated from a strictly mathematical viewpoint if one utilizes the following approach suggested by Szabo and Ostlund.[1]

Assume the existence of a complete set of one-electron functions, ${\chi(x_1)}$, such that any function which depends on the coordinates of that electron can be expanded exactly as

\begin{displaymath}\Phi(x_1) = \sum_i a_i \chi_i(x_1). \end{displaymath} (2)

Now consider the expansion of a function dependent upon the coordinates of two electrons, x1 and x2. If x2 is held fixed, then the function can be written as

\begin{displaymath}\Phi(x_1,x_2) = \sum_i a_i(x_2) \chi_i(x_1), \end{displaymath} (3)

where the expansion coefficient ai is parametrically dependent upon the coordinates of the second electron. However, since the expansion coefficient is itself a function of the coordinates of a single electron, it can also be expanded as

\begin{displaymath}a_i(x_2) = \sum_j b_{ij} \chi_j(x_2), \end{displaymath} (4)

where the choice of the running index and letter designating the expansion coefficient were changed to avoid confusion. Substituting equation (2.4) into the equation for $\Phi(x_1, x_2)$ yields

\begin{displaymath}\Phi(x_1, x_2) = \sum_{ij} b_{ij} \chi_i(x_1) \chi_j(x_2). \end{displaymath} (5)

Requiring that the function $\Phi(x_1, x_2)$ meet the antisymmetry requirement for a wave function describing a system of fermions,

\begin{displaymath}\Phi(x_1,x_2) = - \Phi(x_2,x_1), \end{displaymath} (6)

implies that bij = -bji and bii=0, or

\begin{displaymath}\Phi(x_1,x_2) = \sum_{j>i} [\chi_i(x_1) \chi_j(x_2) - \chi_j(x_1) \chi_i(x_2) ]. \end{displaymath} (7)

With the exception of the normalization constant, the term in square brackets is the expanded form of a Slater determinant for a two-electron system. The above equation may, therefore, be written as

\begin{displaymath}\Phi(x_1,x_2) = \sum_{j>i} 2^{\frac{1}{2}} b_{ij} \vert \chi_i \chi_j \rangle. \end{displaymath} (8)

This demonstrates that any function which depends upon the coordinates of two electrons may be expressed as a linear combination of all unique two-electron Slater determinants. In general, any function depending on the coordinates of N-electrons may be exactly expressed as a linear combination of all possible N-electron Slater determinants formed from a complete set of spin orbitals ${\chi_i(x)}$:

\begin{displaymath}\Psi(x_1, x_2, \cdots x_n) = \sum_i C_i \vert \Phi_i \rangle. \end{displaymath} (9)

The linear expansion coefficients, which also incorporate any normalization issues, are known as the CI coefficients. Having decided that Slater determinants are an appropriate choice of N-particle basis functions, one can proceed with the matrix mechanics solution of the Scrödinger equation by substituting the expansion of the N-electron wave function, equation (2.9), directly into the electronic wave equation:

\begin{displaymath}\sum_i C_i \vert \Phi(x_1 \cdots x_n) \rangle = E \sum_i C_i \vert \Phi(x_1 \cdots x_n) \rangle. \end{displaymath} (10)

Multiplying through on the left by $\mid \Phi_j >$ yields

\begin{displaymath}\sum_i C_i \langle \Phi_j \vert \hat{H} \vert \Phi_i \rangle = E \sum_i C_i \langle \Phi_j \vert \Phi_i \rangle, \end{displaymath} (11)

which may be rewritten as the matrix equation

\begin{displaymath}{\bf H C} = {\bf E S C} \end{displaymath} (12)

provided the elements of the Hamiltonian matrix H are given by $ H_{ij} = \langle \Phi_j \vert \hat{H} \vert \Phi_i \rangle$ and the elements of the overlap matrix by $S_{ij} = \langle \Phi_j \vert \Phi_i \rangle$. The Hamiltonian matrix elements, Hij, may be evaluated using Slater's rules introduced in Chapter 2. Finally, if a set of orthonormal orbitals are used to construct the Slater determinants, then the overlap matrix becomes the identity matrix and equation (2.12) reduces to the eigenvalue equation

\begin{displaymath}{\bf H C}^i = {\bf {E C}^i}. \end{displaymath} (13)

The eigenfunction, Ci, is a vector of CI coefficients which represents the wave function for an electronic state of the molecule in the N-particle basis set; whereas, the associated eigenvalue corresponds to the electronic energy of that state. The subscript i denotes that a number of eigenvalue-eigenvector pairs equal to the dimension of the Hamiltonian matrix exist, with the lowest energy pair describing the ground electronic state.

The question now becomes, ``How may one solve equation (2.13) for the set of eigenvalues and eigenvectors?'' It turns out that the solution of equation (2.13) for the set of eigenvalues and eigenvectors is equivalent to finding the unitary matrix transformation which diagonalizes the Hamiltonian matrix. The columns of the unitary transformation matrix correspond directly to the eigenvectors whereas the diagonal elements of H correspond to molecular energies. It is advantageous to formulate the eigenvalue problem as a matrix diagonalization problem because numerous practical algorithms for diagonalizing matrices have been implemented on modern computers. Two of the most popular methods used by quantum chemists for diagonalizing the Hamiltonian matrix are the Davidson algorithm[2] and the Davidson-Liu algorithm. [3] The advantages of these two methods are that they are iterative and can be used to find only the lowest few roots without requiring the full effort necessary to determine all of the roots. This is significant because the lowest few roots are typically the only ones of chemical interest.

This page maintained by Brian C. Hoffman
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